Zero coupon yield curve

[Oracleyoda]

I have the most basic question.

I have mastered stripping the curve and bootstrapping, I can get the mathematical result and I understand why investment banks create Zeros

What I am a little hazy on is exactly what uses one puts the zero yield curve to.

I am happy to read up, but most of the texts skip this elementary question and jump straight into the method (which of course I get)

Many thanks
Wayne

 

[David Harper CFA FRM]

Hi Wayne,

I agreed the texts (FRM at least) tend to gloss it over; FYI, a good text is (CFA-based) Fabozzi's Fixed Income Analysis
(http://www.amazon.com/Fixed-Income-Analysis-Institute-Investment/dp/047005221X/)

In regard to *valuation,* the primary use of spot rates is to discount cash flows. In theory, each asset's future cash flow ought to be discounted by the zero/spot rate with the exact same maturity; as in theory, a coupon generating bond ought to be decomposed into a series of zero coupon bonds, each cash flow discounted with the corresponding zero rate. The risk-free (default-free) version of the zero rate is the what Fabozzi call's the "theoretical spot rate curve;" i.e., the curve of zero rates that ought to be used to discount future streams of riskless cash flows. (then, of course, spreads can be added to create corresponding risky zero curves, but still with the idea: the rate should vary by the maturity). I think *maybe* there is an underlying premise here, too: that the other yield metrics generally commingle properties of the instrument (e.g., discount bond?, yield to maturity is a function of the instruement) with the term structure. But the zero rate term structure (i think in theory) is directly unobservable but has an existence independent of the instruments; Tuckman's law of one price says something like "there must be a 5% riskless spot rate, that is the same for all 5 year riskless cash flows." And, importantly, the discount factors are a function of the zero/spot rates. So, in the case of the (riskless) theoretical spot rate curve I think the idea is to observe the "objective" term structure that represents the purest consensus view of a discount function (set of discount factors) that is used to account for the time value of money.

In regard to risk, it's more complicated but still: many (most?) methods are based a shift in the zero term structure (duration and duration-related improvements) or shifts in a forward curve extracted from a zero term structure. So, here, i'd suggest bond sensitivity metrics generally use the zero curve as the best-highest "objective" (i.e., independent of the instrument, a macro factor or set of factors) measure of interest rate risk. What i mean is, when we want to stress test interest rate risk (market risk), we don't typically start with a portfolio's yield to maturity or other yield metric (because they comming instrument properties), we generally use something based on the idea of a shift/twist/move in the "objective" term structure of spot rates.

David

 

[Oracleyoda]

David,

This is very useful and much appreciated.

If can extend just a little further.

The texts that I have been using describe the process for constructing forwards, swaps and futures are fairly straight forward. In all of the examples they tend to present the rates as a given.

From your explanation it strikes me that the only valid rates to be used in the construction of any of these instruments (other than say a forward of an instrument with the same characteristics as a given coupon bond) would be the zero spot, as all others, as you say are specific to the instrument.

i.e when they talk about the curve or rates in this context, they are always talking about the spot curve?

Is this correct?

Thank you again

Wayne

 

[David Harper CFA FRM]

Wayne,

I maybe didn't mean to go that far, sorry i don't feel articulate today. I did mean, yes, zero (spot) interest rates are the key building block: there is a direct function between zero rates and the discount factors (e.g., Tuckman) used to determine the present value (price) of a bond.

but Re: "when they talk about the curve or rates in this context, they are always talking about the spot curve?"

No, we cannot make this assumption. A good question will be precise. The term "interest rate" is imprecise against the FRM cirriculum. The term "yield" does connote "yield to maturity (YTM)" but that is neither a spot rate curve nor a forward rate curve. IMO, "term structure" is also imprecise, a good question should specify "zero term structure" or "forward curve" (and even a forward curve needs more information, one year forwards?). Like all of Hull's examples and practice, he is careful to specify (e.g., Treasury zero rates, LIBOR Zero curve). Notice that even zero rate curve is imprecise: we think Treasuries, but Hull is more precise.

Like Hull valuing an interest rate swap, he shows you can do that with a zero curve (treat as two bonds) or with forward curve (FRA). Because the forward is embedded in the spot rate curve. So, neither is the default (in fact, in advanced duration risk, the forward curve is preferred, and would arguably be the default).

So I think "term structure" implies a set of rates but is imprecise until we specify spot/zero term structure or forward term structure
And "yield" defaults to YTM. The exam has tested with only the term "yield" assuming you will know that is YTM.
And, just to remind, even if I say to you "theoretical spot rate curve" is {2% @ 0.5 years, 3% @ 1.0 years...etc}, I have still omitted the information about compound frequency; do i mean continous (Hull) or semi-annual (Tuckman)?

Thanks, David